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Regular ultrafilters and finite square principles. (English) Zbl 1160.03026

Summary: We show that many singular cardinals \(\lambda \) above a strongly compact cardinal have regular ultrafilters \(D\) that violate the finite square principle \(\square ^{\text{fin}}_{\lambda ,D}\) introduced in [J. Kennedy and S. Shelah, J. Symb. Log. 67, No. 3, 1169–1177 (2002; Zbl 1012.03048)]. For such ultrafilters \(D\) and cardinals \(\lambda \) there are models of size \(\lambda \) for which \(M^{\lambda }/D\) is not \(\lambda ^{++}\)-universal and elementarily equivalent models \(M\) and \(N\) of size \(\lambda \) for which \(M^{\lambda }/D\) and \(N^{\lambda }/D\) are non-isomorphic. The question of the existence of such ultrafilters and models was raised in [C. C. Chang and H. J. Keisler, Model theory. 3rd rev. edition. Amsterdam: North-Holland (1990; Zbl 0697.03022)].

MSC:

03E05 Other combinatorial set theory
03C55 Set-theoretic model theory
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References:

[1] DOI: 10.1007/s001530050052 · Zbl 0874.03060
[2] More on regular reduced products 69 pp 1261– (2004)
[3] DOI: 10.1007/BF02764857 · Zbl 0681.03016
[4] DOI: 10.1016/0003-4843(72)90001-0 · Zbl 0257.02035
[5] Model theory (1990)
[6] On regular reduced products 67 pp 1169– (2002)
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